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EXAMPLE
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The table of coefficients of x^k in expansion of e.g.f. -exp(n*x)*log(1 - x)/(1 - x) begins:
n = 0: (0), 1, 3, 11, 50, 274, ...
n = 1: 0, (1), 5, 23, 116, 669, ...
n = 2: 0, 1, (7), 41, 242, 1534, ...
n = 3: 0, 1, 9, (65), 452, 3229, ...
n = 4: 0, 1, 11, 95, (770), 6234, ...
n = 5: 0, 1, 13, 131, 1220, (11149), ...
...
This sequence is the main diagonal of the table.
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MATHEMATICA
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Table[n! SeriesCoefficient[-Exp[n x] Log[1 - x]/(1 - x), {x, 0, n}], {n, 0, 19}]
Table[Sum[n^(n - k) Binomial[n, k] k! HarmonicNumber[k], {k, 1, n}], {n, 0, 19}]
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